A323433 Number of ways to split an integer partition of n into consecutive subsequences of equal length.

1, 1, 3, 5, 10, 14, 25, 34, 54, 74, 109, 146, 211, 276, 381, 501, 675, 871, 1156, 1477, 1926, 2447, 3142, 3957, 5038, 6291, 7918, 9839, 12277, 15148, 18773, 23027, 28333, 34587, 42284, 51357, 62466, 75503, 91344, 109971, 132421, 158755, 190365, 227354, 271511

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Formula

a(n) = Sum_y A000005(k), where the sum is over all integer partitions of n and k is the number of parts.

From Seiichi Manyama, Jan 21 2022: (Start)

G.f.: 1 + Sum_{k>=1} A000005(k) * x^k/Product_{j=1..k} (1-x^j).

G.f.: 1 + Sum_{i>=1} Sum_{j>=1} x^(i*j)/Product_{k=1..i*j} (1-x^k). (End)

a(n) = Sum_{i=1..n} Sum_{j=1..n} A008284(n,i*j). - Ridouane Oudra, Apr 13 2023

Example

The a(5) = 14 split partitions:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Maple

b:= proc(n, i, t) option remember; `if`(n=0 or i=1, numtheory
      [tau](t+n), b(n, i-1, t)+b(n-i, min(n-i, i), t+1))
    end:
a:= n-> `if`(n=0, 1, b(n$2, 0)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 15 2019

Mathematica

Table[Sum[Length[Divisors[Length[ptn]]], {ptn, IntegerPartitions[n]}], {n, 30}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0 || i == 1,
     DivisorSigma[0, t+n], b[n, i-1, t] + b[n-i, Min[n-i, i], t+1]];
a[n_] := If[n == 0, 1, b[n, n, 0]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

Prog

(PARI) my(N=66, x='x+O('x^N)); Vec(1+sum(k=1, N, numdiv(k)*x^k/prod(j=1, k, 1-x^j))) \\ Seiichi Manyama, Jan 21 2022
(PARI) my(N=66, x='x+O('x^N)); Vec(1+sum(i=1, N, sum(j=1, N\i, x^(i*j)/prod(k=1, i*j, 1-x^k)))) \\ Seiichi Manyama, Jan 21 2022

Crossrefs

Cf. A000005, A000219, A008284, A101509, A316245, A319066, A323295, A323300, A323307, A323429, A323434.

Sequence in context: A078411 A137630 A320886 * A220489 A229915 A092269

Adjacent sequences: A323430 A323431 A323432 * A323434 A323435 A323436

Keyword

nonn

Author

Gus Wiseman, Jan 15 2019

Status

approved